It is declared that Torah were given to nations, only Jews were willing to receive, therefore Jews became special persons before HaShem. Accordingly, does HaShem require different responsibility(ies) from Jews than gentiles? What is the special responsibility? Does this mean gentiles are free from these responsibility(ies), such as learning Torah, tikun olam, Kosher and Shabbat, Mitzvot in general, since they don't have Torah?
All human beings are obligated in seven laws, known as the "Seven Noahide Laws". These laws were in place before the giving of the Torah, and go back to Creation.
The laws are (as per Wikipedia):
- Do not deny God.
- Do not blaspheme God.
- Do not murder.
- Do not engage in illicit sexual relations.
- Do not steal.
- Do not eat from a live animal.
- Establish courts/legal system to ensure obedience to said laws.
There are entire groups of people who dedicate their lives to obeying these seven commandments, and they are known as "Bnei Noach". "Bnei Noach" means "sons of Noah" and refers to the fact that since everyone is a descendant of Noah, all humans are obligated in the seven laws, as mentioned previously.
A website in which you can learn more about these laws and how they are practiced by Bnei Noach can be found here. Also, for further reading, see this other Wikipedia page, and this article from Chabad.org.
But you ask if Jews have different responsibilities as opposed to non-Jews. The answer is "yes", we do have much more responsibility. G-d gave us the Torah at Mt. Sinai and obligated us in the observance of 613 commandments, which make up the Torah. Since non-Jews were not given the Torah, they are not obligated to fulfill one commandment of the Torah, unless it is one of the aforementioned seven laws.
Of course, a non-Jew is permitted to perform as many laws from the Torah as he wants, even without conversion, but this is an entirely separate discussion, discussed elsewhere on this site.
I understand there are many people on this site who could answer this question much better than me, but this is only a small introduction. Feel free to ask more questions about the Seven Noahide Laws and the place for non-Jews in Judaism. (We have many questions and answers on this site already which deal with this; you might go and check those out.)